The fact that the aircraft would normally go to the end to turn around and then backtrack to that taxiway, the ATC instruction to expedite would have been in that context. Having said that, ATC would normally make that request after landing. But without an ATC transcript or indeed a CVR transcript, we can speculate until the cows come home.
Just simply using the full width of a runway will prevent this from happening. Many captains in my life have done a 180 ON the runway, initiating by going towards the runway edge before making a swing through the centreline. Never starting from its centreline!